10th International Symposium on Turbulence and Shear Flow Phenomena (TSFP10), Chicago, USA, July, 2017

Shock-bubble Interaction Near a Compliant Tissue-like Material

Shusheng Pan, Stefan Adami, Xiangyu Hu, Nikolaus A. Adams*

Chair of Aerodynamics and Fluid Mechanics,Department of Mechanical Engineering, Technical University of Munich

Boltzmannstraße 15, 85748 Garching, Germany*nikolaus.adams@tum.de

ABSTRACTIn this work, we present numerical simulation results for

shock-induced bubble collapse dynamics near tissue-like compliantgelatin phase. We use a sharp-interface model for multiple materi-als to represent the ambient liquid (water), the non-condensable gasphase (air) and the gelatin phase. Employing multi-resolution tech-niques, we investigate the complex interface dynamics and com-pare the results with experimental data from literature. Our aim isto understand and quantify the mechanisms observed during extra-corporeal shock-wave lithotripsy or sonoporation. Therefore, late-stage dynamics of the bubble collapse and tissue penetration arepresented.

INTRODUCTIONFluid-dynamic interaction mechanisms and processes are es-

sential to biotechnology and biomedicine (Yuan et al. (2015);Rooney (1970)). An important example is kidney-stone lithotripsy,side-effects of which are precursors to many other more recentlyproposed and pursued therapeutical approaches to improve drugdelivery or to cancer treatment (Coussios & Roy (2008); Mi-tragotri (2005)). In medical treatment, Extracorporeal shock wavelithotripsy (ESWL) is a typical way to remove calculi in humanbodies (Lingeman et al. (2009)). The underlying mechanism ofESWL is the shock-bubble interaction subject to various materialswhich leads to stone fragmentation and tissue damage (Kodama &Takayama (1998); Kodama & Tomita (2000); Loske (2010); Calvisiet al. (2008); Freund et al. (2009); Johnsen & Colonius (2009);Kobayashi et al. (2011)). Among the most interesting shock-interaction driven biomedical phenomena is the so-called sonopora-tion where acoustic cavitation of micro bubbles leads to temporarysmall-scale cell-membrane perforations (Fan et al. (2012); Prenticeet al. (2005); Khokhlova et al. (2014); Zhong et al. (2011); Ohlet al. (2006)).

At the core of such processes is the generation of highly lo-calized, bubble collapse-generated shock waves which interact withambient fluid and tissue. This emitted shock wave is stronger thanthe initial impulse with orders of magnitude larger maximum pres-sure and hits the gelatin interface. At the same time, a liquid jettowards the interface is generated, induced by the vortical motionof the asymmetric bubble collapse. Both effects can cause a ruptureof the tissue layer, however, the precise mechanisms are unclear andmotivate our detailed numerical simulations.

The potential of such extremely small scale yet high-energyevents enables in situ control of therapeutical fluid processes withhigh precision and minimum side effects. The eventual objective ofthe project of which we present here first numerical investigationsis how shock interactions in life organisms can be harnessed forinnovative nanoscale processes.

In this paper we present first results for a generic configura-tion, resembling the basic mechanism of cell-membrane porationby shock-wave impact, where a gas bubble collapse near a com-pliant wall is initiated by the impact of a planar shock wave. The

compliant wall is modeled as a gel-like fluid which is believed to berepresentative for cell-membrane material. Such setups have beenstudied experimentally e.g. by Kodama & Tomita (2000), here wepresent first results employing state-of-the-art conservative multi-material sharp-interface methods (Hu et al., 2006; Han et al., 2014;Pan et al., 2017).

PHYSICAL MODEL AND NUMERICAL METHODOL-OGY

A bubble filled with non-condensable gas is placed next toa tissue-like material and is subjected to a shock wave in water.A mixture of water and 10% solid gelatin (Kodama & Takayama(1998); Kodama & Tomita (2000)) is employed to mimic the tissue-like material, whose acoustic impedance is 1.62× 106 kgm−2 s−1,which is similar to that of many human organs, e.g., liver or kid-ney (Goss et al. (1978)). The water surrounding the gas bubble alsohas an acoustic impedance of 1.62× 106 kgm−2 s−1, indicating allwaves are transmitted through the water-tissue interface without re-flections. The material properties are listed in Table 1.

Table 1: Initial conditions and material properties.

Materials γ B ρ P u

Air 1.4 0 1.2 P0 0

Gelatin 4.04 6.1×108 1061 P0 0

Post-shock water 4.4 6×108 998.6 P0 0

Pre-shock water 4.4 6×108 ρs Ps us

The configuration is considered axi-symmetric and the stiff-ened equation-of-state is used for all materials. The computationaldomain has a length of 4R0 in the radial direction (x) and 80R0 in theaxis direction (z), where R0 = 0.8mm is the bubble radius. The ini-tial shock wave is located at z= 60R0, 1.4R0 upstream of the bubblecenter. Its overpressure, Ps, ranges from 10.2 MPa to 163.2 MPa.The lower bound is chosen to be the same with that in an experi-ment (Kodama & Takayama (1998)) and the upper bound resemblespressure level observed in typical ESWL treatments Loske (2010).The exponential pressure profile, P(z) = (Ps−P0)eb(z−60R0) +P0,is the same with that in Kobayashi et al. (2011) and correspondsto a laser generated shock in the experiment (Kodama & Takayama(1998)). The velocity and density profile of the shock is determinedby Rankine-Hugoniot relation, see us and ρs in Table 1. The sur-face of the tissue-like material is placed at z = 62.4R0 to make sureit has attached to the downstream pole of the bubble. The chosenmaterial properties, initial conditions and computational domain areconsistent with the experiment setup (Kodama & Takayama (1998);

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Kodama & Tomita (2000)) where the shock wave is generated bylaser focusing and the bubble is attached to the gelatin surface us-ing a syringe.

The Reynolds number and Weber number at the beginning ofpenetration in our shock-driven bubble collapse are Re ∼ O(105)and We ∼ O(107), respectively. Thus, viscous effects and surfacetension effects are neglegcted for the tissue penetration dynamics.Other mechanisms with smaller magnitude than viscous dissipation,such as thermal diffusivity, mass diffusivity and phase change, canalso be neglected (Johnsen & Colonius (2009)). For weak shock-waves with overpressure below the elastic limit, the elastic effect isinitially marginal and dominates only after very large penetration.For overpressures beyond the elastic limit, the elastic effect is com-pletely negligible. Thus we can consider the tissue penetration asa pure inertial process and treat all three materials as immisciblecompressible fluids whose dynamics can be solved with our multi-material sharp interface method (Pan et al. (2017)).

RESULTS AND DISCUSSIONSVerification and Validation

To demonstrate the validation of our numerical method andcomputational setup, we first compare our result with the experi-mental data for a weak shock (Ps/P0 = 102). As shown in Fig.1(a), good agreement is found for moderate penetration depths upto Lp ' 2R0. This supports the assumption, that the penetrationprocess is dominated by inertial effects for small and moderate de-formation. At later times, the penetration decelerates significantlydue to elastic effects.

Note, for large overpressures Ps/P0 = O(103), which are morerealistic in ESWL, we expect more accurate results with our methodas elastic effects are definitely negligible in this regime.

t(µs)

l p(t)

/R0

300 400 500

10-1

100

101

Num

Figure 1: Comparison of simulated penetration dynamicswith experimental data (Kodama & Takayama (1998); Ko-dama & Tomita (2000)) for Ps/P0 = 102. Note, time in thesimulation was adjusted to synchronize with the experiment.

Numerical convergence is verified by monitoring the tempo-ral evolution of the equivalent bubble radius R(t), i.e. the equiva-lent radius of a spherical bubble with the instantaneous gas bubblevolume. Figure 2 shows the effective bubble radius over time dur-ing the collapse with Ps/P0 = 102 for three different resolutions.Clearly, finest resolution shows converged results.

t(µs)

R(t

)/R

0

0 10 20 30 40

10-1

100

101

∆

Figure 2: Equivalent bubble radius over time during a collapsesimulation with Ps/P0 = 102 for different grid resolutions.

t

R(t

)/R

0

10-1 100

10-1

100

101

Ps/P0=102Ps/P0=204Ps/P0=408Ps/P0=816Ps/P0=1632

Collapse Penetration

Figure 3: Time history of equivalent bubble radius during theshock-bubble interaction near a tissue-like material for var-ious overpressures Ps/P0. The dashed lines show the timeinstant of the collapse time tc and onset of tissue penetration.

Bubble collapse behaviourThe most distinct behaviour during the early stage of a ESWL

is the shock-driven bubble collapse. The initial planar shock wavefirst hits the upstream pole of the gas bubble, generating a trans-mitted shock and a reflected rarefaction wave with different propa-gation directions, simultaneously. Then, the upstream interface ofthe bubble begins to deform. The high post-shock pressure in thewater induces contraction of the gas bubble. The transient bubblecollapse is shown in Fig. 3, where the equivalent bubble radius isplotted over time for various overpressures. After the strongly ac-celerated collapse at tc, the bubble expands continuously. For smalloverpressures Ps/P0, the bubble reaches its minimal volume beforepenetrating the gelatin phase. With increasing Ps/P0, the onset ofpenetration occurs earlier and eventually happens before the col-lapse.

The Rayleigh collapse time of a air bubble is estimated bytRc = 0.915

√ρL/(Ps−P0)R0 (Brennen (2013)) for spherical col-

lapse in free-field and tR,∗c = tRc (1+ 0.205R0/H) (Vogel & Lauter-born (1988); Rattray (1951)) for non-spherical collapse near a wall,

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Ps/P0

t

0 500 1000 1500

0

Figure 4: The bubble collapse time, tc, as a function of pres-sure ratio Ps/P0.

where H is the stand-off distance. Due to the presence of the wall,water filling the void region generated by the collapsing bubble ismissing, thus collapse is retarded (Johnsen & Colonius (2009)).As for a rigid-wall boundary, the bubble collapse near a tissue-like material is expected much slower as compared to the free-fieldcollapse. However, given the missing “pressure doubling” effect(wave reflection at rigid wall) since acoustic impedances are iden-tial across the tissue interface, bubble collapse in the current setupis expected even slower as compared to the non-spherical collapsenear a wall. In Fig. 4, the collapse time, tc and the onset of pene-tration, tp, are shown for several pressure amplitudes. Interestingly,the collapse times are comparable to tc = 1.7tRc , which demonstratesthe slow-down of the collapse by the tissue material. Note that thecollapse time for Ps/P0 = 102 is tc = 13.3µs, which agrees very wellwith the experimental finding of tc = 13µs for Ps = 10.2 (Kodama& Takayama (1998) (Fig. 13)). Additionally, we plotted the starttimes of penetration in Fig. 4. As mentioned before, at very strongshockwaves the bubble penetrates the tissue already prior collapse.

Liquid jet, shock formation and tissue surface mi-gration

This tissue surface initially migrates towards the gas bubbledue to the sink flow generated by the collapsing bubble. With in-creasing overpressure of the initial shockwave, this migration issmaller as the collapse time is decreasing and the sink flow ef-fect becomes less significant. However, the migration itself is self-similar over the entire range of pressure amplitudes, see Fig. 5.Here, tissue interface migration is plotted over the rescaled pre-collapse time (t− tc)cL. The numerical data for the different over-pressures can be fitted by a single quadratic function.

Usually a non-spherical bubble collapse will form a re-entrantjet due to the velocity difference of the upstream and downstreamparts, uu−ud (Blake & Gibson (1987)). For a shock-driven collapsenear boundaries, the velocity uu increases with Ps and yields largerelative velocities uu − ud , indicating a strong liquid jet in shockpropagation direction. This high-speed re-entrant jet will gener-ate a water-hammer shock which propagates radially. In turn, thejet hits the tissue surface and increases the surface pressure signif-icantly, as shown in Fig. 6. This pressure satisfies a fitting curve,Pw,max = ρLc2L[logw(Ps/P0)− 2]/10. This is of interest when mea-suring the potential damage of the tissue during the shock-drivenbubble collapse. Note, the spatial pressure profiles along the tissue

+

+

+

+

+

+

*

*

*

(t-tc)c

L

Lm/R

0

-1.5 -1 -0.5 0

0

0.2

0.4

0.6

Ps/P

0=102

Ps/P

0=204

Ps/P

0=408

Ps/P

0=816

Ps/P

0=1632

t = 3.6 (Lm/R

0)2/c

L+t

c

+

*

Figure 5: Migration of tissue surface during bubble collapse.

interface decrease with x (Johnsen & Colonius (2008, 2009)).

Ps/P0

Pw

,max

/ρLc

L2

101 102

0

0.1

0.2

0.3

Figure 6: Maximum pressure on tissue surface at x = 0 asfunction of Ps/P0.

The dynamics of tissue penetrationPenetration of a continuous phase can occur when a high-speed

liquid jet impacts a material, see e.g. Uth & Deshpande (2013), whoinvestigated the unsteady water jet impact on a translucent gel ex-perimentally. For ESWL, the shock-driven penetration is directlyrelated to the potential damage and its dynamics remain largely un-explained.

A whole penetration process of a shock-driven bubble collapsenear a tissue material is illustrated by the interface structures in Fig.7. For larger Ps/P0, the penetration increases and more and moregas is convected into the tissue material. Simultaneously, the bub-ble elongation in vertical direction increases. The enlarged subfig-ure shows that the width of the liquid jet also increased with Pssignificantly, which is consistent with shock-induced collapse neara wall (Johnsen & Colonius (2009)). Consequently, the amount ofliquid perfusing the tissue phase increases.

Fig. 8 shows the transient tissue penetration after collapse fordifferent overpressures Ps/P0. Except for late times the results are

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10.9μs

2mm

22.8μs 34.8μs

8.5μs 17.1μs 32.0μs

5.2μs 12.7μs 30.8μs

3.2μs 7.6μs

28.9μs

2.2μs 6.6μs

18.1μs

A

PS = 10.2MPa

B

PS = 20.4MPa

C

PS = 40.8MPa

D

PS = 81.6MPa

E

PS = 163.2MPa

Liquid jet

Liquid jet

Figure 7: Snapshots of interface topology before bubble col-lapse (left), for small penetration depth (middle) and largepenetration depth (right). The regions colored by white, yel-low and blue denote air, tissue and water, respectively. Theliquid jet front is enlarged to compare its thickness for differ-ent Ps.

self-similar. We derived an analytical relation for the penetrationdepth Lp(t) as function of time using mass and momentum conser-vation. The resulting scaling law is given by

Lp(t)R0∼

(√PsρG

ρ2G−ρ2L

tR0

) 23

∼(

t√

Ps√ρLR0

) 23

∼(

ttRc

) 23

. (1)

This correlation is plotted together with the simulation resultsin Fig. 8. The scaling law is valid from t∗p = 0.2 to 4.5 and indicatesa deceleration of the penetration rate. Note, the scaled penetrationtime is defined as t∗p = t/

√ρL/Ps/R0. For late times t∗p > 4.5 the

behavior differs as in this regime the dynamics are significantly af-fected by elastic forces.

Non-attached bubble collapse and penetrationSo far we have studied gas bubbles attached to the gelatin in-

terface. Now, we present results for a non-attached shock-drivenbubble collapse near a tissue-like material. The stand-off distance,H, is set to 1.2R0 as in Kobayashi et al. (2011). The qualitative flowevolution with bubble collapse and tissue penetration is shown inFig. 9. The snapshots clearly show the bubble collapse, the emittedspherical shock wave due to the water hammer event, the re-entrantliquid jet formation and and the tissue penetration. Notice the mul-tiple wave reflections and complex pressure wave pattern due to a

+

+

+

+

+

++

++

++++

++++

+++++++

++++++

+++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++

*

*

****

**

***

****

******

*************

**************************************************************

(t-tc)/(ρ

L/P

s)0.5

/R0

Lp/R

0

10-1

100

101

10-1

100

101

Ps/P

0=102

Ps/P

0=204

Ps/P

0=408

Ps/P

0=816

Ps/P

0=1632

+

*

[(t-tc)/t

c

R]2/3

[(t-tc)/t

c

R]1.0

Figure 8: The penetration-time behaviour for different pres-sure ratio, Ps/P0 = 102 ∼ 1632. The scaling law is given toprovide a simple model for predicting the dynamics of shock-driven tissue penetration. The time t is recorded from initialcondition, not the penetration instant.

highly irregular bubble deformation. A grid convergence study forthe temporal evolution of the equivalent bubble radius is shown inFig. 10. The converged collapse time for this case is 1.6tRc , whichis in good agreement with the numerical result of Kobayashi et al.(2011) and agrees with the fitting in Fig. 4. Also, the scaled pene-tration depth still exhibits the same scaling law, (t/tRc )

2/3, see Fig.11.

CONCLUSIONSWe have presented numerical simulations of shock-bubble in-

teractions near a tissue-like gelatin phase using a compressiblemulti-material sharp-interface model. Numerical convergence wasverified by grid refinement studies. We have shown that dependingon the strength of the initial shockwave, the bubble can penetratethe tissue-like gelatin phase already prior maximal compression. Atthe same time, the strength and width of the liquid jet increases andleads to larger ambient fluid entrainment in the compliant material.A quantitative comparison of the penetration depth with experimen-tal data from literature shows very good agreement. We found auniversal scaling law for the penetration depth as a function of post-collapse time. This relation holds also for the experimental data, yetwe seek for more profound validation with additional references.The numerical simulations give a detailed insight into the bubble-collapse and penetration dynamics. Different to experiments, verylocalized quantitative data can be extracted to understand the phys-ical mechanisms that lead to various penetration scenarios as uti-lized for treatments like ESWL. More profound investigations ofthe liquid-jet in combination with more complex material modelscan help to understand and improve e.g. drug delivery into cells andare in the focus of our current research.

ACKNOWLEDGMENTThis project has received funding from China Scholarship

Council (No. 201306290030) and the European Research Coun-cil (ERC) under the European Union’s Horizon 2020 research andinnovation programme (grant agreement No 667483). Parts of theresults in this work were obtained during the CTR Summer Program2016. The authors gratefully acknowledge the Gauss Centre for Su-percomputing e.V. for funding this project by providing computing

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Figure 9: Bubble collapse, shock formation and tissue penetration for a non-attached bubble near a tissue-like material.

t/tcR

R(t

)/R

0

0.5 1 1.5 2 2.5 3

0.2

0.4

0.6

0.8

1

∆=R0/25

∆=R0/26

∆=R0/27

∆=R0/28

∆=R0/29

Figure 10: Equivalent bubble radius over time for a non-attached bubble with a stand-off distance H = 1.2R0 andPs/P0 = 1080.

time on the GCS Supercomputer SuperMUC at Leibniz Supercom-puting Centre.

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